Homology, Homotopy and Applications

Volume 20 (2018)

Number 2

Distributed computation of low-dimensional cup products

Pages: 41 – 59

DOI: http://dx.doi.org/10.4310/HHA.2018.v20.n2.a3

Authors

Nisreen Alokbi (School of Mathematics, National University of Ireland, Galway, Ireland; and Department of Mathematics, College of Computer Science and Mathematics, University of Wasit, Iraq)

Graham Ellis (School of Mathematics, National University of Ireland, Galway, Ireland)

Abstract

We describe a distributed algorithm for computing the cup product $\cup : H^1 (X, \mathbb{Z}) \times H^1 (X, \mathbb{Z}) \to H^2(X, \mathbb{Z})$ on the cohomology of a finite regular CW-space. A serial implementation of the algorithm is illustrated in two applied topological settings: (i) 3-dimensional digital images; (ii) topological data analysis of a finite sample of points from a metric space. For the second of these illustrations we introduce a cohomological enrichment of the Mapper clustering procedure which may be of independent interest.

Keywords

cohomology, cup product, digital image, data analysis

2010 Mathematics Subject Classification

55N99

Full Text (PDF format)

Received 8 May 2017

Received revised 16 December 2017

Published 18 April 2018